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International Journal of Soft Computing and Engineering (IJSCE) ISSN: 2231-2307, Volume-5 Issue-1, March 2015

A Deterministic Inventory Model For Weibull Deteriorating Items with Selling Price Dependent Demand And Parabolic Time Varying Holding Cost Vipin Kumar, Anupama Sharma, C.B.Gupta They classified inventory models on the basis of demand variations and various other conditions or constraints. With exponential declining demand and partial backlogging, Ouyang and Cheng [2005] developed an inventory model for deteriorating items. Alamri and Balkhi[2007] studied the effects of learning and forgetting on the optimal production lot size for deteriorating items with time-varying demand and deterioration rates. Dye [2007] find an optimal selling price and lot size with a varying rate of deterioration and exponential partial backlogging. They assume that a fraction of customers who backlog their orders increases exponentially as the waiting time for the next replenishment decreases. Roy [2008] developed a deterministic inventory model when the deterioration rate is time proportional. Demand rate is a function of selling price, and holding cost is time dependent. Liao [2008] gave an economic order quantity (EOQ) model with non instantaneous receipt and exponential deteriorating item under two level trade credits. Pareek et al. [2009] developed a deterministic inventory model for deteriorating items with salvage value and shortages. Skouri et al. [2009] developed an inventory model with ramp-type demand rate, partial backlogging, and Weibull's deterioration rate. Mishra and Singh [2010] developed a deteriorating inventory model for waiting time partial backlogging when demand and deterioration rate is constant. Tripathy and Mishra [2010] give the model for deteriorating items with price dependent demand and linear holding cost. Mandal [2010] gave an EOQ inventory model for Weibull-distributed deteriorating items under ramp-type demand and shortages. Hung [2011] gave an inventory model with generalized-type demand, deterioration, and backorder rates. Mishra and Singh [2011] gave an inventory model for ramp-type demand, timedependent deteriorating items with salvage value and shortages and deteriorating inventory model for timedependent demand and holding cost with partial backlogging. In this paper, we extend the paper of Tripathy, Mishra[2010]. we developed generalized EOQ model for deteriorating items where deterioration rate follows twoparameter Weibull distribution and holding cost are expressed as parabolic functions of time and demand rate considered to be function of selling price. For the model where shortages are allowed they are completely backlogged. Here we have considered both the case of with shortage and without shortage in developing the model.

Abstract- This paper with development of an inventory model when deterioration rate follows Weibull two way parameter distributions. It is assumed that demand rate is function of selling price and holding cost is parabolic in terms of time. In this models both the cases with shortage and without shortage are taken into consideration. Whenever shortage allowed is completely backlogged. To illustrate the result numerical examples are given .the sensitive analysis for the model has been performed to study the effect of changes the value of parameters associated with the model. Mathematics Subject Classification: 90B05 Keywords: - EOQ model, deteriorating items, Weibull distribution, shortage, price dependent demand, parabolic holding cost.

I.

Introduction

These days researcher are paying more attention to the inventory model of deteriorating items. It’s not easy to neglect the effect of deterioration, as it is a realistic feature and is very common in daily routine life. So have to consider it. Wee HM [1993] is the first one who define deteriorating items refers to the items that become decayed, damaged, evaporative, expired, invalid, devaluation and so on by the passing time. Traditionally, it was considered that the items can preserve their characteristics while they kept stored in inventory. But it is not true for all. Considering this fact, now a days it’s a great challenge to control and maintain the inventory of deteriorating items for the decision makers. The first EOQ inventory model was developed by Harris [1915], which was further generalized by Wilson [1934] to obtain formula for economic order quantity. Within [1957] studied the deterioration of the fashion goods at the end of the prescribed shortage period. Ghare and Schrader [1963] developed a model for an exponentially decaying inventory. Mishra [1975] develop a model with Weibull deterioration rate without backordering. Moving further, Dave and Patel [1981] were the first to study a deteriorating inventory with linear increasing demand when shortages are not allowed. This model was further generalized by Sachan[1984] with allowed shortages. Further, work in this field has been done by Chung and Ting [1993]; Wee [1995] studied an inventory model with deteriorating items. Chang and Dye [1999] developed an inventory model with time-varying demand and partial backlogging. Goyal and Giri[2001] gave recent trends of modeling in deteriorating item inventory.

II.

Manuscript Received on February 2015. Vipin Kumar, Dept. of Mathematics BKBIET, Pilani Anupama Sharma, Research Scholar Mewar University, Rajasthan, India C.B. Gupta, Dept. of Mathematics BITS, Pilani

Assumptions and Notations

The fundamental assumptions of this model are as follows:a)

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The deterioration rate is proportional to time Published By: Blue Eyes Intelligence Engineering & Sciences Publication Pvt. Ltd.

A Deterministic Inventory Model For Weibull Deteriorating Items with Selling Price Dependent Demand And Parabolic Time Varying Holding Cost b) The deterioration of units follows the two parameter Weibull distribution

θ (t ) = αβ t

g) T is the completelength of cycle. h) Replenishment is instantaneous and lead time is zero. i) The order quantity in one cycle is q. j) A is the cost of placing an order. k) The selling price per unit item is p. l) C1 is the unit cost of an item. m) The inventory holding cost per unit per unit time is h(t). n) C2 is the shortage cost per unit per unit time.. o) Inventory is depleted due to deterioration and demand of the item. At time t1 the inventory becomes zero and shortage starts occurring.

β −1

where 00. Selling price p follows an increasing trend, demand rate possess the negative derivative throughout its domain where demand rate is f ( p ) = ( a − p ) >0

Mathematical formulation and solutions :Let Q(t) be the inventory level at time t ( 0 ≤ t are given by

dQ (t ) + αβ t β −1Q (t ) = −( a − p ) dt dQ (t ) = −(a − p ) dt With

≤ T ). The differential equations to describe instantaneous state over (0,T)

0 ≤ t ≤ t1

……… (1)

t1 ≤ t ≤ T

……….(2)

= 0 =

Solving equation (1) and equation (2) and neglecting higher power of α, we get

  α β +1 Q (t ) = ( a − p )(1 − αt β ) (t1 − t ) + (t1 − t β +1 )  β +1  

0 ≤ t ≤ t1

And

Q(t ) = (a − p)(t1 − t )

t1 ≤ t ≤ T

Now stock loss due to deterioration is given by:t

D = ∫ αβ t β −1Q(t )dt 0

αt β +1  = (a − p) 1   β +1 

…..(3)

Ordering quantity is given by:T

q = D + ∫ (a − p)dt 0

q = (a − p)

αt β +1 + (a − p)T β +1

αt β +1  q = (a − p)  +T  β +1 

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Published By: Blue Eyes Intelligence Engineering & Sciences Publication Pvt. Ltd.

International Journal of Soft Computing and Engineering (IJSCE) ISSN: 2231-2307, Volume-5 Issue-1, March 2015 Holding costis :t

H = ∫ (h + δt 2 )Q(t )dt 0

 β  ∫ (h + δt )(a − p )(1 − αt )(t t1

=

2

1

− t) +

0

ℎ ℎ

Neglecting higher power of

t

H= h( a − p ) 

β +2

2 1

 2

+

α β +1

(t1

β +1

 − t β +1 ) dt 

, we get

 α β t1  t1 4  α 2 t1 αβ t1 β + 4 α 2 t1 2 β + 4 + −  δ a p ( − ) + −   …..(4) 2 ( β + 1)( β + 2) 2( β + 1)   12 3( β + 3)( β + 4) ( β + 3)(2 β + 4)  2β +2

Now Shortage cost during the cycle:T

S = ∫ Q(t )dt t1

S = (a − p)

(t1 − T ) 2 2

……..(5)

From equation (3), (4) and (5). Total profit per unit time is given by P(T,t1,p)= p(a-p)-

1 (A+ C1q+ H + C2S) T

   α t β +1 + ( a − p )T  +  A + C 1 (a − p ) 1 β +1     =p(a-p)2 αβ t 1 β + 2 α 2 β t1 2 β + 2 1    t (a − p )h 1 + −  T ( β + 1)( β + 2 ) 2 ( β + 1) 2   2   1  C 2  ( a − p )( t 1 − T ) 2   2  

    4 β +4 2β +4 2  t    αβ t 1 α t1 +δ 1 +  − +   12 3 ( β + 3 )( β + 4 ) ( β + 3 )( 2 β + 4 )          

……….(6) Lett1= T 0 < γ < 1     αγ β +1T β +1 + ( a − p )T  +  A + C 1 ( a − p )  β +1     P(T,p)=p(a-p) - 1   hγ 2 T 2 hαβγ β + 2 T β + 2 hα 2 γ 2 β + 2 T 2 β + 2 δT 4 γ 4 δαβ T β + 4 γ β + 4 δα 2 T 2 β + 4 γ 2 β + 4   ( a − p )  + − + + −  + 2 β β β β β β T  2 ( + 1 )( + 2 ) 12 3 ( + 3 )( + 4 ) ( + 3 )( 2 + 4 ) β + 2 ( 1 )      C  1 (a − p )T 2 (γ − 1)2     2  2    

………..(7)

This equation gives the profit function P(T,p). In order to maximize the profit function P(T,p). the necessary conditions for are given by

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A Deterministic Inventory Model For Weibull Deteriorating Items with Selling Price Dependent Demand And Parabolic Time Varying Holding Cost

∂P(T , p ) ∂P (T , p ) =0 = 0 and ∂T ∂p We get  hγ 2 hβγ β + 2T β +1 hα 2 (2 β + 1)T 2 β γ 2 β + 2 3δT 2γ 4 + − +  ( β + 2) ( β + 1) 2 12  (a − p )αγ β +1β T β −1  A  2 − 2 + C1  + ( a − p )  2  δα ( β + 2) β + 2 β + 4 T ( β + 1) δα (2 β + 3) 2 β + 2 2 β + 4   T γ − T γ  ( β + 3)(2 β + 4)  ( β + 4)

……..

 + 1 2   +C2  ( a − p )(1 − γ )  =0 2     

(8)

And

  αγ β +1T β +1   −T + C1 −  β +1      β +4 β +4 2 2 β +2 β +2 2 2β +2 2β +2 4 4 2 2β +4 2β +4 (a-2p)- 1      α ( β + 2)T γ α T γ α βγ δT γ T  =0 − h γ T + αβγ T − + − − δ    T   2 ( β + 1)( β + 2) 12   ( β + 3)( β + 4) ( β + 3)(2β + 4)  ( β + 1) 2   − C 2 (T 2 − T 2 γ 2 )     

{

}

……… (9)

Using the software Mathematica-5.1 and Ms. Excel from the equations (8) and (9) we can calculate the optimum value of T* and P* simultaneously and the optimal value P*(T,P) of the average net profit is determined by (7) provided they satisfy

∂ 2 P(T , P) ∂ 2 P(T , P) thesufficiency conditions for maximizing P (T,P) are